A formal proof of Sylow’s theoremAn experiment in abstract algebra with Isabelle Hol

Florian Kammüller, Lawrence C. Paulson

November 1998, 30 pages

Abstract

The theorem of Sylow is proved in Isabelle HOL. We follow the proof by
Wielandt that is more general than the original and uses a non-trivial
combinatorial identity. The mathematical proof is explained in some
detail leading on to the mechanization of group theory and the necessary
combinatorics in Isabelle. We present the mechanization of the proof in
detail giving reference to theorems contained in an appendix. Some weak
points of the experiment with respect to a natural treatment of abstract
algebraic reasoning give rise to a discussion of the use of module
systems to represent abstract algebra in theorem provers. Drawing from
that, we present tentative ideas for further research into a section
concept for Isabelle.